Feature EMS Newsletter June 2017 7 Andrew Wiles’ Marvellous Proof * Henri Darmon (McGill University, CICMA and CRM, Montreal, Canada) Fermat famously claimed to have discovered “a truly marvel- lous proof” of his Last Theorem, which the margin of his copy of Diophantus’ Arithmetica was too narrow to contain. While this proof (if it ever existed) is lost to posterity, Andrew Wiles’ marvellous proof has been public for over two decades and has now earned him the Abel prize. According to the prize ci- tation, Wiles merits this recognition “for his stunning proof of Fermat’s Last Theorem by way of the modularity conjecture for semistable elliptic curves, opening a new era in number theory”. Few can remain insensitive to the allure of Fermat’s Last Theorem, a riddle with roots in the mathematics of ancient Greece, simple enough to be understood and appreciated by a novice (like the 10-year-old Andrew Wiles browsing the shelves of his local public library), yet eluding the concerted efforts of the most brilliant minds for well over three cen- turies. It became, over its long history, the object of lu- crative awards like the Wolfskehl prize and, more impor- tantly, it motivated a cascade of fundamental discoveries: Fer- mat’s method of infinite descent, Kummer’s theory of ideals, the ABC conjecture, Frey’s approach to ternary diophantine equations, Serre’s conjecture on mod p Galois representa- tions, . . . Even without its seemingly serendipitous connection to Fermat’s Last Theorem, Wiles’ modularity theorem is a fun- damental statement about elliptic curves (as evidenced, for instance, by the key role it plays in the proof of Theorem 2 of Karl Rubin’s contribution to the issue of the Notices of the AMS mentioned above). It is also a centrepiece of the “Langlands programme”, the imposing, ambitious edifice of results and conjectures that has come to dominate the number theorist’s view of the world. This programme has been de- scribed as a “grand unified theory” of mathematics. Taking a Norwegian perspective, it connects the objects that occur in the works of Niels Hendrik Abel, such as elliptic curves and their associated abelian integrals and Galois representations, with (frequently infinite-dimensional) linear representations of the continuous transformation groups, the study of which was pioneered by Sophus Lie. This report focuses on the role of Wiles’ Theorem and its “marvellous proof” in the Lang- lands programme, in order to justify the closing phrase in the prize citation: how Wiles’ proof has opened “a new era in number theory” and continues to have a profound and lasting impact on mathematics. Our “beginner’s tour” of the Langlands programme will only give a partial and undoubtedly biased glimpse of the full panorama, reflecting the author’s shortcomings as well as the inherent limitations of a treatment aimed at a general reader- * This report is a very slightly expanded transcript of the Abel prize lecture delivered by the author on 25 May 2016 at the University of Oslo. It is published with the permission of the Notices of the AMS: reprinted from Volume 64, Issue 3, March 2017. ship. We will motivate the Langlands programme by starting with a discussion of diophantine equations: for the purposes of this exposition, they are equations of the form X : P( x 1 ,..., x n+1 ) = 0, (1) where P is a polynomial in the variables x 1 ,..., x n+1 with in- teger (or sometimes rational) coefficients. One can examine the set, denoted X(F), of solutions of (1) with coordinates in any ring F. As we shall see, the subject draws much of its fascination from the deep and subtle ways in which the behaviours of different solution sets can resonate with each other, even if the sets X(Z) or X(Q) of integer and rational solutions are foremost in our minds. Examples of diophan- tine equations include Fermat’s equation x d + y d = z d and the Brahmagupta-Pell equation x 2 - Dy 2 = 1 with D > 0, as well as elliptic curve equations of the form y 2 = x 3 + ax + b, in which a and b are rational parameters, the solutions ( x, y) with rational coordinates being the object of interest in the latter case. It can be instructive to approach a diophantine equation by first studying its solutions over simpler rings, such as the complete fields of real or complex numbers. The set Z/nZ := {0, 1,..., n - 1} (2) of remainders after division by an integer n ≥ 2, equipped with its natural laws of addition, subtraction and multiplica- tion, is another particularly simple collection of numbers, of finite cardinality. If n = p is prime, this ring is even a field: it comes equipped with an operation of division by non-zero el- ements, just like the more familiar collections of rational, real and complex numbers. The fact that F p := Z/ pZ is a field is an algebraic characterisation of the primes that forms the ba- sis for most known efficient primality tests and factorisation algorithms. One of the great contributions of Evariste Galois, in addition to the eponymous theory that plays such a crucial role in Wiles’ work, is his discovery of a field of cardinality p r for any prime power p r . This field, denoted F p r and some- times referred to as the Galois field with p r elements, is even unique up to isomorphism. For a diophantine equation X as in (1), the most basic invariant of the set X(F p r ):=  ( x 1 ,..., x n+1 ) ∈ F n+1 p r such that P( x 1 ,..., x n+1 ) = 0  (3) of solutions over F p r is of course its cardinality N p r := #X(F p r ). (4) What patterns (if any) are satisfied by the sequence N p , N p 2 , N p 3 ,..., N p r ,...? (5) Andrew Wiles’ Marvellous Proof * Henri Darmon (McGill University, Montreal, Canada)